On page 9 of The Handbook of the Geometry of Banach Spaces: Volume I I found the following:
"A basis $\{x_n\}_{n=1}^{\infty}$ is said to be an unconditional basis provided that $\sum \alpha_n x_n$ converges unconditionally whenever it converges. This is equivalent to saying that every permutation of $\{x_n\}_{n=1}^{\infty}$ is also a basis."
How are those two statements equivalent?
Use the following:
Let $X$ be a Banach spacce.
1) $(y_n)$ is a basis of $X$ if and only if each $x\in X$ has a unique representation $\sum\limits_{i=1}^\infty \alpha_i y_i$.
2) If $\sum\limits_{i=1}^\infty z_i$ converges unconditionally to $z$, then any rearangement of $\sum\limits_{i=1}^\infty z_i$ converges to $z$.
3) If $(x_n)$ is a basis of $X$ with coefficient functionals $(x_n^*)$ and if $x=\sum\limits_{i=1}^\infty \alpha_i x_{\pi(i)}$, then $x_{\pi(i)}^* x= \alpha_i$.